Math 442: Differential Geometry of Curves and Surfaces

David Dumas

University of Illinois at Chicago
Fall 2010

General information

CRN	32336 (undergraduate), 32337 (graduate)
Lectures	MWF 11:00 - 11:50am in Stevenson 215
Text	Differential Geometry of Curves and Surfaces by Manfredo do Carmo (see also: list of errata) ISBN-13: 978-0-13-212589-5
Instructor	David Dumas (ddumas@math.uic.edu)
Office hours	Mondays and Wednesdays 2-3pm in SEO 503
Exam week office hours	Monday, Dec 6, 1-4pm
Homework Policies	Homework problems will be assigned on Mondays and Wednesdays, and occasionally on Fridays.
	Homework is due by 4pm on the first lecture day of each week (usually Monday) and can be turned in during lecture or directly to the grader's mailbox (Jonah Gaster) in the mailroom on the 3rd floor of SEO.
	The list of challenge problems (see course materials) will be updated on a regular basis; you must complete four of these problems during the semester, with at least two of them submitted before the midterm exam.
	Late homework will not be accepted.
	You can discuss the homework with other students, but you must write (and understand!) your own solutions.
	The homework problems depend on material from both the assigned reading and the lectures. Do not leave the problem sets or reading to the last minute!

Course Materials

These documents are available for download in PDF format.

• Course Syllabus
• List of errata in the textbook, compiled by Bjorn Poonen and his students.
• Challenge problems - Last updated
• Crofton's formula example and Crofton matching exercise
• Midterm topic outline
• Midterm exam and solutions
• Final exam topic outline
• Formula reference sheet that will be provided during the final exam

Important Dates

There will be one midterm exam and a cumulative final exam.

Curve Day	Wed, Sep 22	SEO 612
Midterm Exam	Wed, Oct 13	SH 215
Surface Day	Wed, Nov 24	SEO 612
Final Exam	Thurs, Dec 9, 10:30am-12:30pm	SH 215

Grading

Your course grade will be determined on the following basis:

Homework	30%
Midterm Exam	30%
Final Exam	40%

When computing your homework grade, the lowest weekly homework score will be dropped, and the remaining scores will be averaged with your challenge problem scores. (That is, each challenge problem is worth as much as a weekly homework assignment!)

Assignments

Reading

Unless otherwise noted, all reading assignments refer to the textbook, Differential Geometry of Curves and Surfaces by Manfredo do Carmo.

Homework

Online Resources

These online tools (java applets, etc.) complement the material from the textbook and lectures.

• Integral geometry lecture notes by Andrejs Treibergs
  • See also: Inequalitites that imply the isoperimetric inequality, which includes the proof we will discuss in lecture
• Curvature flow demonstrations
  • Curvature flow java applet by James Sethian (Berkeley) - draw your own curve, watch its evolution.
  • Curve shortening flash demo by Sigurd Angenent (University of Wisconsin) - high-resolution demonstration of the evolution of a complicated simple curve.
• 3d-XplorMath: Demonstration software for curves and surfaces in space
  • Some of the demonstrations are also available as java applets which you can view in your web browser.
• JavaView: Interactive 3D Geometry and Visualization
  • Geodesic surveyor - compute geodesics on (polyhedral) surfaces
  • Model viewer - view and manipulate polyhedral models
• Caustics in Differential Geometry by Oliver Knill and Michael Teodorescu, an HCRP project that includes:
  • Geodesics on a torus of revolution
  • Geodesics on the surface of revolution of a Gaussian
• Frenet frame applet by Martin Raussen at Aalborg University. This applet shows the Frenet frame of an arbitrary parametric curve (by default, the twisted cubic).
• Make your own paper model of the great dodecahedron, by Gijs Korthals Altes

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